Best response polytopes - exercises

1. Define a polytope as a convex hull
2. Define a polytope as an intersection of halfspaces
3. For the following set of vertices draw the corresponding polytope and obtain their definition as an intersection of halfspaces:
   1. V = $\{(0, 0), (0, 1), (1, 0), (1, 1)\}$
   2. V = $\{(0, 0), (0, 1/4), (1, 2/3), (2, 1/5)\}$
   3. V = $\{(0, 0), (0, 1/4), (1, 2/3), (2, 1/5), (1, 0)\}$
4. Define the best response polytopes.
5. For the following games, obtain the best response polytopes, label the vertices and identify all Nash equilibria:
   1. $ A = \begin{pmatrix} 3 & -1\\ 2 & 7\end{pmatrix} \qquad B = \begin{pmatrix} -3 & 1\\ 1 & -6\end{pmatrix} $
   2. $ A = \begin{pmatrix} 2 & -1\\ 1 & 3\end{pmatrix} \qquad B = \begin{pmatrix} -2 & 2\\ 1 & -2\end{pmatrix} $